Properties and Postulates

Date: 08/04/99 at 11:23:40
From: Kiki Nwasokwa
Subject: Properties and Postulates
When people discover (or create) a property, do they just discover it
ONCE and then know that from then on it applies to all similar
situations, or do they just happen to keep discovering the property
until they decide to call it a property? In other words, how long does
it take for something to become a property?
COMMUTATIVE PROPERTY:
1. The commutative property of addition seems so intuitive and
fundamental - (obviously if you have an apple, an orange, and a lemon
in a box, you could also call them a lemon, an apple, and an orange
and still be describing the same box) - that it is almost like a
postulate. What distinguishes postulates from properties such as the
commutative property of addition?
2. How can people be sure that properties such as the commutative
property of multiplication, which are not as intuitive as the
commutative property of addition, work in every case? (I think I found
a way to prove this property - but I want to know if it is something
that even NEEDS to be proven.)

Date: 08/20/99 at 12:46:30
From: Doctor Ian
Subject: Re: Properties and Postulates
Hi Kiki,
In theory, once you've discovered a property - that is, proved that
some theorem is true - then you never need to discover it again. And
anyone who is playing the same game as you (for example, standard
number theory) can use your discovery to make more discoveries of his
own.
But that's in theory. In practice, you would have to publish your
result, and other people would have to read and verify it.
Gauss made several major discoveries that he wrote in his diary, and
which only became known decades after his death, after some of them
had been discovered independently by other mathematicians.
Similarly, Isaac Newton invented the Calculus in order to prove to his
own satisfaction that an inverse square law of force would result in
an elliptical orbit, but he didn't tell anyone until Edmund Halley
asked him about it many years later. In Germany, Leibniz, not knowing
what Newton had done, invented it on his own, using a different
notation.
The Indian mathematician Ramanujan 'discovered' many results by some
intuitive process that no one understood, but which clearly didn't
involve the notion of 'proof'. So while he was able to report
interesting discoveries, some turned out to be wrong, others had to be
proven by mathematicians who couldn't have dreamed them up, and some
remain unproven today.
But these are all exceptional cases. Usually, a property becomes known
when a mathematician either observes or guesses that some kind of
pattern exists, proves that it does, tells other mathematicians about
it, and has his proof verified by independent mathematicians. At that
point, other mathematicians can treat it as if it were 'obviously'
true.
> The commutative property of addition seems so intuitive and
> fundamental... that it is almost like a postulate. What distinguishes
> postulates from properties such as the commutative property of
> addition?
You're right that many properties seem so basic that it's tempting to
think of them as postulates. But one of the primary differences
between postulates and properties is that the number of postulates in
a given formal system either stays the same or decreases, while the
number of properties continues to grow.
Properties are knowledge, and knowledge is power, so you want to have
as many properties as you can find.
But postulates are assumptions, so you want to have as few of them as
you can get away with.
That's why, for centuries, mathematicians tried to 'prove' Euclid's
parallel postulate using the other postulates as a starting point. And
that's why mathematicians find it preferable to prove things like the
commutative property of addition, even though from a certain point of
view, proving something so obvious seems like a waste of time.
>How can people be sure that properties such as the commutative
>property of multiplication, which are not as intuitive as the
>commutative property of addition, work in every case?
There are two ways to know that something applies in every case. The
first is to make it a postulate; the second is to prove it as a
theorem. If you rule out divine inspiration, those are really your
only choices. Since the commutative properties of addition and
multiplication aren't postulates, they do need to be proven.
By the way, I don't agree with you that the commutative property of
multiplication is any less 'intuitive' than the commutative property
of addition. Visually, you can represent a sum of two numbers like
this:
+--+---+
|**|***|
+--+---+
Flip it around, and you get
+---+--+
|***|**|
+---+--+
Since it's the same object, the order of the operation can't matter.
Similarly, you can represent the product of two numbers like this:
* * * * * *
* * * * * *
* * * * * *
Rotate it 90 degrees, and you get
* * *
* * *
* * *
* * *
* * *
Again, since it's the same object, the order of the operation can't
matter.
But it's important to remember that a picture isn't a proof. A picture
shows that something is true for one particular case, while a proof
shows that it is true for all possible cases. When you want to
convince yourself of something, a picture is often good enough. But
when you want to prove it to someone else - especially someone who
might be using it as a starting point for discoveries of his own - you
have to meet a higher standard.
Also, if you remember how Russell's paradox led to a re-examination of
the foundations of mathematics, you'll see that we often learn the
most by trying to really understand the simplest cases, rather than
the more complicated ones.
I hope this helps. Be sure to write back if you have other questions.
- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/